Parallel systems, Random variables, Stochastic orders
Let X-lambda 1, X-lambda 2, ... ,X-lambda n be independent non negative random variables with X-lambda i similar to F(lambda(i)t), i = 1, ... , n, where lambda(i) > 0, i = 1, ... , n and F is an absolutely continuous distribution. It is shown that, under some conditions, one largest order statistic X-n:n(lambda) n is smaller than another one X-n:n(theta) according to likelihood ratio ordering. Furthermore, we apply these results when F is a generalized gamma distribution which includes Weibull, gamma and exponential random variables as special cases.
Subhash C. Kochar and Nuria Torrado, On stochastic comparisons of largest order statistics in the scale model, to appear in Communications in Statistics - Theory and Methods.